Curve generator for generating a smooth curve with a pair of circular arcs between two given points

ABSTRACT

A curve generator for generating a smooth curve with a pair of circular arcs between the two given points P 1  and P 2  on the condition that the generated curve is tangent with the given tangent lines M 1  and M 2  at said points P 1  and P 2 , comprises means for defining the coordinates of the point Q of the connection of said pair of arcs on the locus satisfying the formula 
     
         ∠P.sub.1 QP.sub.2 =π-1/2(θ.sub.1 +θ.sub.2) 
    
     where θ 1  and θ 2  are angles between the line P 1  P 2  and the tangent lines M 1  and M 2  respectively. Either the major arc or minor arc between the points P 1  and Q, and between the points Q and P 2 , that is to say, the revolutional direction of each arc is selected. Thus, a pair of arcs P 1  Q and QP 2  connect the two given points P 1  and P 2  smoothly.

BACKGROUND OF THE INVENTION

The present invention relates to a curve generator, in particular, relates to a curve generator for generating a plurality of smooth arcs in a graphic display device.

A prior graphic display device displays an approximate curve by generating a plurality of short linear lines or short vectors. However, said prior art has the disadvantages that the data or the short vectors required are very voluminous, thus the amount of memory for storing said data and the duration of time for displaying and/or transmitting the curve are very large. Further, the curve thus obtained by the approximation of a plurality of short vectors is not sufficiently smooth. To obtain a smoother curve, more data is required.

Other prior curve generators are, for instance, an analog integrator type curve generator (Timothy E. Johnson; "Analog Generator for Real-Time Display of Curves", Technical Report No-398, Lincoln Laboratory, MIT, 1965), and a digital curve generator with the combination of digital differential analyzers (James R. Armstrong; "Design of a Graphic Generator for Remote Terminal Application", IEEE Transactions for Computers, Vol. C-22, No. 5, 1973). However, those prior arts have the disadvantages that the structure or the circuit is very complicated and only a very limited curve can be generated by those curve generators, and thus those prior arts are not practicable.

Another prior art attempted to display a curve with a plurality of arcs. However, according to a prior arc type curve generator, a profile of a curve is defined uniquely by the coordinates of the whole given points and the tangential line at the start point, thus the curves thus generated have the undesired oscillatory phenomenon as shown in FIG. 1, in which the dotted line shows an ideal curve. That oscillatory phenomenon is discussed in detail in "An approximation of a curve with circular arcs", by T. Kamae and M. Kosugi, Information Processing Society of Japan, Vol 12, 1972. If an oscillatory phenomenon occurs, the generated curve oscillates as shown in FIG. 1, although it is smooth in a short section. Accordingly, a curve generator which generates a curve using a plurality of arcs has not been implemented in spite of the simple structure of same.

Other prior curve generators are U.S. Pat. No. 3,860,805, U.S. Pat. No. 3,325,630, and British Pat. No. 866,319. Those prior arts generate a curve between two points using a plurality of parabola or linear lines, and have the disadvantages that the structure of the curve generator is complicated and the curve thus generated is not sufficiently smooth.

Another prior art for generating a curve between two points is "Biarc Curves" by D. M. Bolton, (British Ship Research Association), Computer Aided Design, Vol 7, No. 2, 1975 (British publication). Although that document suggestes a curve generator which connects two points by a pair of arcs, that document does not refer to the locus of the junction point of the two arcs. Therefore, the flexibility of a curve thus generated is very limited, and further the calculation for obtaining a curve is rather complicated resulting a complicated device.

SUMMARY OF THE INVENTION

It is an object of the present invention, therefore, to overcome the disadvantages and limitations of prior curve generators by providing a new and improved curve generator.

The present curve generator for fulfilling the above object provides a curve having a pair of arcs connecting two given points P₁ and P₂ smoothly on the condition that the generated curve is tangent with the given lines M₁ and M₂ passing through said points P₁ and P₂, and the present curve generator comprises means for defining the coordinates of the junction point Q of said pair of arcs on the locus satisfying the formula

    ∠P.sub.1 QP.sub.2 =π-1/2(θ.sub.1 +θ.sub.2),

where θ₁ and θ₂ are angles between the line P₁ P₂ and the tangent lines M₁ and M₂ respectively; means for deciding the revolutional direction of each arc, and means for generating an arc according to the output of said two means.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features, and attendant advantages of the present invention will be appreciated as the same become better understood by means of the following description and the accompanying drawings wherein;

FIG. 1 is the explanatory drawing showing the oscillatory phenomenon when a single arc is used between each adjacent two points in a prior art,

FIG. 2 shows a curve connecting two points by a pair of arcs satisfying the tangent lines at said two points, according to the present invention,

FIG. 3 is the explanatory drawing showing the mathematical analysis of the locus of the junction point of a pair of arcs according to the present invention,

FIG. 4 is the example of a curve connecting two points by a pair of arcs,

FIG. 5 shows a group of possible curves each connecting two points by a pair of arcs according to the present invention,

FIG. 6 is a block-diagram of a curve generator according to the present invention,

FIG. 7 is a detailed block-diagram showing the curve processor 4 in FIG. 6,

FIG. 8 is the explanatory drawing showing the decision of the proper revolutional direction of an arc according to the present invention,

FIG. 9 is a block-diagram of a slope calculator 15 in FIG. 7,

FIG. 10 is a block-diagram of an m₁ calculator 17 in FIG. 7,

FIG. 11 is a block-diagram of an m₂ calculator 18 in FIG. 7,

FIG. 12 is a block-diagram of the unit in the m₂ calculator 18 in FIG. 11,

FIG. 13 is a block-diagram of a junction point calculator (Q calculator) 20 in FIG. 7,

FIG. 14 is a block-diagram of a direction circuit 28 in FIG. 7,

FIG. 15, FIG. 16 and FIG. 17 are block-diagrams of a subtractor, a multiplier and a divider, respectively, used in the above calculators.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

First, the mathematical principle that when the coordinates of the two points and a pair of tangent lines at those points are given, a curve with a pair of arcs can connect said two points smoothly, will be explained in accordance with FIG. 2 to facilitate understanding of the present invention.

In FIG. 2 the two points are designated P₁ and P₂, the tangent lines at those points are M₁ and M₂, and the angles between the line P₁ P₂ and the tangent lines M₁, M₂ are θ₁ and θ₂, respectively. In the above designation, the curve with a pair of arcs C₁ and C₂, which are tangential with the tangent line M₁ at point P₁ and the tangent line M₂ at point P₂, respectively, can connect those points P₁ and P₂. That is to say, when the junction point Q of the arcs C₁ and C₂ is defined, the tangent line of the arc C₁ at point Q can be the same as the tangent line of the arc C₂ at point Q. And further, the point Q is not a stationary point, but can move on the locus.

FIG. 3 shows the mathematical analysis of said locus of the junction points of a pair of arcs. In FIG. 3, it is supposed that ∠QP₁ P₂ =α, ∠QP₂ P₁ =β, the cross points of the tangent line at point Q and the tangent lines M₁, M₂ are R and S, respectively. Since the formula below

    ∠RQP.sub.1 +∠P.sub.1 QP.sub.2 +∠P.sub.2 QS=π

is satisfied, the other formula below is introduced from the above formula.

    ∠P.sub.1 QP.sub.2 =π-1/2(θ.sub.1 +θ.sub.2)=(const)

That is to say, the point Q resides on the locus of the constant circle K on which the angle of Q for the line P₁ P₂ is constant.

The center O of the circular arc K is on the perpendicular bisector of the line P₁ P₂, and ∠P₁ OP₂ =θ₁ +θ₂ is satisfied, as shown in FIG. 3. Also, α+β=1/2 (θ₁ +θ₂) is satisfied. Accordingly, as apparent from FIG. 4, a pair of arcs C₁ (P₁ Q) and C₂ (QP₂) can be connected smoothly using the point Q. In FIG. 4, m₁ is the slope of the line P₁ Q, m₂ is the slope of the line QP₂, M is the slope of the line P₁ P₂, and γ is the angle ∠P₂ P₁ X.

It should be noted that there can be infinite number of curves which have said arcs C₁ and C₂ passing through the point Q, since the point Q can move the entire length from the point P₁ to the point P₂ on the locus K.

FIG. 5 shows some examples of the generated curves according to the present invention. The curves having a pair of arcs C₁ (i) (i=1˜8) and C₂ (i) (i=1˜8) passing through the point Q₁ (i=1˜8). It is supposed in FIG. 5 that θ₁ =30° , θ₂ =60°, α_(i) =∠Q_(i) P₁ P₂ =(9-i/9) (θ₁ +θ₂)/2=5 (9-i), i=1, 2, . . . , 8. It is one of the features of the present invention that a plurality of curves can be generated according to the value of the parameter α. Further, it should be noted that the sign of the curvature of the arcs C₁ and C₂ is not defined, therefore, not only a convex curve between the points P₁ and P₂, but also a curve having an inflection point between the points P₁ and P.sub. 2 can be possible. The latter curve can also satisfy the condition of the tangential lines M₁ and M₂ at the points P₁ and P₂, respectively, by using the point Q on the locus K.

FIG. 6 is the block-diagram of the curve generator according to the present invention. In FIG. 6, the reference numeral 1 is a display control unit, 2 is a memory for storing the display data, 3 is a curve decoder, 4 is a curve processor which is the most important unit in the present invention, 5 is an arc generator, 6 is a display unit with a Cathode-Ray-Tube. The display control unit 1 receives the display data from the memory 2, and the curve decoder 3 decodes the curve display instruction from the display data and applys the data necessary for the generation of a curve to the curve processor 4. Said data necessary for the generation of a curve is, in the present embodiment, the coordinates of the point P₁ (X₁, Y₁), the coordinates of the point P₂ (X₂, Y₂), the slope of the tangent line M₁ at the point P₁, the slope of the tangent line M₂ at the point P₂, the value of tan (θ₁ +θ₂)/2, and the value of tan α which defines the location of the point Q on the locus K. It should be appreciated that the circuit of the curve generator will be quite simplified in structure when α and β are designed to be θ₁ /2 and θ₂ 12, respectively.

The curve processor 4 processes the above data and generates first the data for the first arc and applys the same to the arc generator 5, then the arc generator 5 generates the first arc and displays the same on the screen of the display unit 6. Next, the display unit receives the data for displaying the second arc from the curve processor 4 and displays the second arc on the screen. Thus the curve having a pair of curves connected smoothly is displayed on the screen.

FIG. 7 is the block-diagram showing in detail the structure of the curve processor 4. In FIG. 7, the reference numeral 11 is a start point register, 12 is an end point register, 13 is a start point tangent register, 14 is an end point tangent register, 15 is a slope calculator, 16 is a delay circuit, 17 is an m₁ calculator, 18 is an m₂ calculator, 19 is a delay circuit, 20 is a Q calculator, 21 is a delay circuit, 22, 23, 24 and 25 are selectors, 26 is an OR-circuit, 27 is a delay circuit, 28 is a direction circuit, 29 is a delay circuit, 30 is a binary-counter, 31 is a single-shot circuit. Further, the reference numeral 41 is a register for storing the value tan (θ₁ +θ₂)/2, 42 is a variable resistor, and 43 is an analog-digital convertor.

In the present invention, it is supposed that the point Q on the locus K is specified by defining the value α (or tan α) in FIGS. 3 or 4. Said parameter tan α is directly applied to the curve processor 4. Of course said parameter tan α can also be applied from the decoder 3. Further, the value tan (θ₁ +θ₂)/2 which is used for the calculation of m₂ could be calculated in the curve processor 4 from the values M₁ and M₂, although said value is applied to the curve processor 4 from the decoder 3 in the present embodiment.

As explained above, the coordinates of the start point P₁ are (X₁, Y₁), the coordinates of the end point P₂ are (X₂, Y₂), the tangent lines at the points P₁ and P₂ are M₁ and M₂ respectively, and the angles between the tangent lines M₁ and M₂ and the line P₁ P₂ are θ₁ and θ₂, respectively.

The curve processor 4 receives the data necessary for the calculation of the curve from the decoder 3 when the curve display instruction is decoded in the decoder 3. That is to say, the register 11 receives the coordinates (X₁, Y₁) of the point P₁, the register 12 receives the coordinates (X₂, Y₂) of the point P₂, the register 13 receives the tangent M₁ at the point P₁, the register 14 receives the tangent M₂ at the point P₂, and the register 41 receives the value (θ₁ +θ₂). Further, the rotational angle of the variable resistor 42 provides the value v=tan α, which is converted to a digital value by the digital-analog converter 43.

When all the data mentioned above are applied to the curve processor 4, the start pulse (START) is applied to the curve processor 4 from the decoder 3. The start pulse (START) triggers first the slope calculator 15, which then calculates the slope M of the line P₁ P₂ according to the equation

    M=(Y.sub.2 -Y.sub.1)/(X.sub.2 -X.sub.1)

using the coordinates of the points P₁ and P₂ in the registers 11 and 12.

Next, the start pulse (START) triggers the m₁ calculator 17 and the m₂ calculator 18 through the delay circuit 16, where m₁ and m₂ are slopes of the lines P₁ Q, and QP₂ in FIG. 4, respectively. First, the calculation of the value m₁ will be explained. The formula for the calculation of m₁ is shown below.

    m.sub.1 =tan (γ+α)=(M+tan α)/(1-Mtan α)

As the angle between the tangent line M₁ and the line P₁ P₂ is θ₁, and the angle between the tangent line M₂ and the line P₁ P₂ is θ₂, then the value tan γ which is the slope M of the line P₁ P₂ in FIG. 4 (line P₁ X shows the reference direction of an angle) provides the above formula. Accordingly, the m₁ calculator 17 calculates the value m₁ from the values M and v=tan α. The detailed structure of the m₁ calculator 17 will be shown later.

On the other hand, the tangent m₂ of the line QP₂ is obtained as follows.

    m.sub.2 =tan (γ-β)=(M-tan β)/(1+Mtan β)

Since α+β=1/2(θ₁ +θ₂), mentioned before, the value of tan β is

    tan β=(T-v)/(1+Tv)

where v=tan α, and T=tan (θ₁ +θ₂)/2.

Accordingly, the m₂ calculator 18 receives the value v from the analog-digital convertor 43, and T=tan (θ₁ +θ₂)/2 from the register 41, then calculates tan β first, and next calculates the value m₂ using said tan β and the value M from the slope calculator 15. The detail of the m₂ calculator 18 will be shown later.

The values m₁ and m₂ thus calculated are stored in the registers (not shown) in the m₁ calculator 17 and the m₂ calculator 18 respectively. It should be appreciated of course that the other calculators in FIG. 7 have registers for storing the calculation result although they are not shown for the sake of simplicity.

Next, the start pulse (START) which triggered the m₁ calculator 17 and the m₂ calculator 18 triggers the Q calculator 20 through the delay circuit 19. Then, the Q calculator 20 receives the coordinates of the points P₁ and P₂, and the tangents m₁ and m₂, then calculates the coordinates of the point Q (X_(q), Y_(q)) which is the cross point of the line m₁ and the line m₂, in the following formula.

    X.sub.q =(Y.sub.2 -Y.sub.1 +m.sub.1 X.sub.1 -m.sub.2 X.sub.2)/(m.sub.1 -m.sub.2)

    Y.sub.q =(m.sub.1 Y.sub.2 -m.sub.2 Y.sub.1 +m.sub.1 m.sub.2 X.sub.1 -m.sub.1 m.sub.2 X.sub.2)/(m.sub.1 -m.sub.2)

When the coordinates of the junction point Q(X_(q), Y_(q)) are calculated said start pulse (START) controls the selectors 22, 23, 24 and 25 through the delay circuit 21 so that those selectors provide the data for the first arc (the arc C₁ in FIG. 4). The start pulse also clears the binary counter 30 to "00". Those selectors include the registers for storing the above data, and the integrated circuit for constructing the selector is for instance SN7429 produced by Texas Instruments Co., Ltd. The selector 23 stores the coordinates of the start point of the first arc P₁ (X₁, Y₁), and the selector 24 stores the coordinates of the end point of the first arc Q(X_(q), Y_(q)), the selector 22 stores the data of the tangent line M₁ at the point P₁ of the first arc, and the selector 25 stores the data m₁ which defines the revolutional direction of the first arc or selects either the major arc or minor arc for the first arc from the entire circle. Next, the start pulse (START) triggers the direction circuit 28 through the OR-circuit 26 and the delay circuit 27. The direction circuit 28 thus triggered receives the value m₁ from the selector 25, and the value M₁ from the selector 22, then defines the revolutional direction of the first arc. When it is assumed that the length of the arc is less than a semi-circle as is generally used, the sign of the value D calculated throuth the following formula can define the revolutional direction of the arc.

    D=(M.sub.1 -m.sub.1)/(1+M.sub.1 m.sub.1)

That is to say, when the sign of the value D is positive, the arc rotates in the clock-wise direction, and when the sign of the value D is negative, the arc rotates in the anti-clock-wise direction. FIG. 8 explains the above situation, that is, when angle difference, Δθ=M₁ -m₁ is defined, the sign of the value Δθ coincides with the sign of tan Δθ so long as |Δθ|<(π/2) is satisfied. However, please note that the inverse of the sign described later is not performed in the direction circuit 28 when the selectors 22, 23, 24, 25 select the data for the first arc. Although the above embodiment presumes that the length of the arc is less than a semi-circle, the longer arc can be available. When the arc is greater than a semi-circle, the values of ΔX, and ΔY of the tangent line, instead of the tangent value of the tangent line, as the data of the tangent line, can be used for calculating the revolutional direction of the arc.

Now, the data for the first arc are provided to the arc generator 5 from the curve processor 4 and said start pulse (START) triggers the arc generator 5.

The arc generator 5 generates the arc according to the data supplied to the same. An arc generator is well known and the arc generator 5 can be any one of those well known arc generators. In some cases, an arc generator requires the coordinates of the start point of the arc, the coordinates of the end point of the arc, the coordinates of the center of the arc, and the revolutional direction of the arc (for instance, Vector General Inc, CAG--1 type arc generator), as the input date to the arc generator, instead of the data explained in FIG. 7. In that case, the coordinates of the center of the arc (X₀, Y₀) are calculated from the output of the apparatus in FIG. 7 through the following formula. ##EQU1## where i=1 or 2 designating first or second arc,

    X.sub.mi =(X.sub.i +X.sub.q)/2

    Y.sub.mi =(Y.sub.i +Y.sub.q)/2

    m.sub.i =(Y.sub.q -Y.sub.i)/(X.sub.q -X.sub.i)

It should be appreciated that the data conversion as explained above for the interface of the curve processor with an arc generator is obvious to those skilled in the art.

When the generation of the first arc is finished, the arc generator 5 provides the end pulse, which changes the content of the binary counter 30 to "01", so that the less significant digit of the binary counter 30 allows the selectors 22, 23, 24, 25 to provide the data for the second arc (C₂), through the single-shot multivibrator 31. That is to say, the selectors in that situation, provide the coordinates of the start point Q, the coordinates of the end point P₂, the tangent line M₂ at the point P₂, and the slope m₂ of the line QP₂.

The operation for generating the second arc is the same as that for the first arc except for the revolutional direction (note; D₂ =(M₂ -m₂)/(1+M₂ m₂)). That is to say, the selectors 22, 23, 24 and 25 provide the data for the second arc to the arc generator 5, which generates the second arc. However, since the tangent M₂ at the end point is used for defining the revolutional direction of the second arc, when the sign of the difference m₂ -M₂ is positive, the arc rotates in the clock-wise direction, and when the difference is negative, the arc rotates in the anti-clock-wise direction. Accordingly, the direction circuit 28 must change the sign for the second arc. Thus, when the start pulse (START) switches the selectors 22˜25 to the second arc, said start pulse (START) changes the sign of the output of the direction circuit 28. The detailed structure of the direction circuit 28 will be shown later.

When the second arc is generated, the arc generator 5 provides the end pulse, which changes the content of the binary counter 30 to "10". In this case the single-shot circuit 31 is not triggered and the pulses to the selectors 22, 23, 24 and 25 are inhibited. Further, the output of the binary counter 30 is applied to the "READY" terminal, which informs the curve decoder that the next curve display can be prepared. Thus, the complete cycle for generating a curve having a pair of arcs connected smoothly has finished. As apparent from the above explanation, when a train of points and the tangent at each points are given, a curve segment having a pair of arcs is produced between each adjacent two points, and thus all the given points are connected by a continuous smooth curve having a plurality of arcs.

Now, the detailed structure of the members 15, 17, 18, 20 and 28 which are shown in FIG. 7 will be described.

FIG. 9 shows the embodiment of the slope calculator 15, which is composed of a pair of subtractors (SUB) 101 and 102 for producing Y₂ -Y₁ and X₂ -X₁, respectively, and a divider 103 for calculating (Y₂ -Y₁)/(X₂ -X₁) from the output of said subtractors. In FIG. 9, the upper input line and the lower input line of the subtractors (SUB) 101 and 102 receive the minuend (Y₂, X₂), and the subtrahend (Y₁, X₁), respectively, and the upper line and the lower line of the divider 103 receive the dividend and the divisor respectively. The detailed structure of the subtractor and the divider will be shown later.

FIG. 10 shows the structure of the m₁ calculator 17 which is composed of a full adder (FA) 104, a multiplier (MUL) 105, a subtractor (SUB) 106, and a divider (DIV) 107. In the m₁ calculator 17, the dividend of the divider 107 is M+v (=M+tan α), which is provided by the full adder (FA) 104, and the divisor of the divider 107 is 1-Mv(=1-Mtan α), which is provided by the multiplier (MUL) 105 and the subtractor (SUB) 106, thus the divider 107 provides the quotient (M+v)/(1-Mv). The embodiment of the full adder 104 is for instance the integrated circuit SN7483 produced by Texas Instrument Co, Ltd. The structure of the multiplier 105 will be shown in later.

FIG. 11 is the detailed block-diagram of the m₂ calculator 18, which comprises a pair of calculators 108 and 109. Each calculator calculates the formula (A-B)/(1+AB) where A and B are variables. The first calculator 108 receives the value T(=tan (θ₁ +θ₂)/2), and v(=tan α), and provides the output tan β (=(T-v)/(1+Tv)). The second calculator 109 receives the tan β thus obtained and the value M, and provides the value of m₂ =(M-tan β)/(1+Mtan β).

FIG. 12 shows the detailed block-diagram of the calculators 108 and 109 in FIG. 11. The calculators 108 and 109 are composed of a subtractor (SUB) 110, a multiplier (MUL) 111, a full adder (FA) 112, and a divider (DIV) 113. The divider 113 receives the value T-v (or M-tan β) from the subtractor (SUB) 110 as dividend, and the value 1+Tv (or 1+Mtan β) from the multiplier (MUL) 111 and the full adder (FA) 112 as divisor. Thus, the divider 113 provides the quotient tan β=(T-v)/(1+Tv) , or m₂ =(M-tan β)/(1+Mtan β).

FIG. 13 shows the detailed embodiment of the Q calculator 20, which is composed of six multipliers (MUL) 114, 115, 117, 118, 119 and 120, three subtractors (SUB) 116, 125 and 126, four full adders (FA) 121, 122, 123, 124, and two dividers (DIV) 127 and 128. The Q calculator 20 receives the input signals m₁, m₂, X₁, Y₁, X₂ and Y₂. The divider (DIV) 127 receives the value m₁ X₁ +Y₂ -m₂ X₂ -Y₁ as dividend, and the value m₁ -m₂ as divisor, and said divider (DIV) 127 provides the quotient ##EQU2## of the X coordinate of the point Q.

Similarly, the divider (DIV) 128 receives the value m₁ m₂ X₁ +m₁ Y₂ -m₁ m₂ X₂ -m₂ Y₁ as the dividend, and the value m₁ -m₂ as the divisor, then provides the quotient of the Y coordinate ##EQU3## of the point Q. It should be appreciated that the structure of the subtractors, the multipliers and the dividers is completely the same as that used in the m₁ calculator 17 or m₂ calculator 18.

FIG. 14 shows the block-diagram of the direction circuit 28, which comprises the calculator 129 for the calculation of (A-B)/(1+AB) shown in FIG. 12, a flip-flop 130, inverters 131 and 132, AND circuits 133 and 134, and OR circuit 135. In selecting the revolutional direction of the first arc, the calculator 129 receives the values M₁ and m₁ from the selectors 22 and 25, and then the calculator 129 calculates (M₁ -m₁)/(1+M₁ m₁) and the sign of the same. Further, the binary counter 30 (FIG. 7) applys the information as to whether the arc is the first or the second one to the direction circuit 28. When the arc is the first arc, the output of the flip-flop 130 is zero., therefore, the output of the calculator 129 is directly provided to an outside circuit without changing the sign. On the other hand when the arc is the second arc, the values M₂ and m₂ are applied to the direction circuit 29 from the selectors 22 and 25 (FIG. 7), and the calculator 129 calculates the value (M₂ -m₂)/(1+M₂ m₂) and the sign of the same. And further the output of the flip-flop 130 is one "1" during the second arc concerns, and thus the output of the calculator 129 is inverted by passing through the inverter 131, and the inverted output is provided to an outside circuit.

Now, the structure of the subtractor (SUB) 101, 102, 106, 110, 116, 125 and 126, the multiplier (MUL) 105, 111, 114, 115, 117, 118, 119 and 120, the divider (DIV) 103, 107, 113 127 and 128 will be explained.

FIG. 15 shows the block-diagram of the subtractor (SUB), which comprises a full adder (FA) 140 and a plurality of inverters. The minuend A is directly applied to the full adder (FA) 140, while the subtrahend B is applied to the full adder (FA) 140 through the inverters, together with the carry. The inverters and the carry change the value B to the 2's complement of the same. Thus, the full adder (FA) 140 provides the difference A-B, instead of A+B.

FIG. 16 shows the embodiment of the multiplier (MUL), which comprises the integrated circuit 141 for multiplier, for instance, Am 25LS14 produced by Advanced Micro Devices Co, Ltd, a parallel-in-serial-out shift register 142, a serial-in-parallel-out shift register 143. The shift registers 142 and 143 can be, for instance, the integrated circuits SN74165, and SN74164, produced by Texas Instrument Co, Ltd, respectively. The multiplicand A and the multiplier B can be applied to the present multiplier in a parallel form. Since the integrated circuit 141 for the multiplier is designed to receive the mutliplier in a serial form, a clock signal CLK is necessary for converting the parallel input signal to the serial input signal. It should be appreciated that many integrated circuits for multiplication other than Am 25LS14 are available on the market.

FIG. 17 shows the embodiment of the divider (DIV), which is based on a non-restoring method. The divider (DIV) comprises a full adder (FA) 150, a pair of shift registers (SR) 151 and 152, a flip-flop (F/F) 153, and provides the quotient A/B from the dividend A and the divisor B. The integrated circuit SN74198 produced by Texas Instrument Co, Ltd, can be used as said shift registers 151 and 152. In FIG. 17, the exclusive OR circuit 156 determines whether the sign of the divisor coincides with the sign of a partial remainder, and if they coincide the flip-flop 153 is reset to zero, accomplishing the subtraction and the bit of the quotient is set to one. On the other hand, if the sign of the divisor does not coincide with the sign of the partial remainder, the addition is accomplished and the bit of the quotient is set to zero. Next, the divisor is shifted by one bit. The operation of the division is accomplished by repeating the above cycle.

As explained above in detail, the present invention can provide a smooth curve with a pair of arcs between two points, which could have been connected with only a Spline curve or a high order polynominal in a prior art. Further, in a prior art, a Spline curve generator could not be realized due to the complex structure of the same, a curve has been approximated with a plurality of short linear lines, and so a considerable amount of data has been required in a prior art. On the other hand, since the present invention uses only a pair of simple arcs, the necessary data for generating a curve is merely 1/10 to 1/50 of the necessary data in a prior art. Further, the generated curve according to the present invention is smoother than that of a prior art.

Accordingly, the present invention can be used in a CRT graphic display unit, an X-Y plotter, and/or a numerical control (NC) system, which displays, draws or cuts a smooth curve. Further, the present invention has the effect that the amount of the memory in a computer system can be reduced and the transmission time for sending or receiving the information concerning a curve can be shortened, since the amount of data for defining a curve is reduced in the present invention.

From the foregoing it will now be apparent that a new and improved curve generator has been found. It should be understood of course that the embodiments disclosed are merely illustrative and are not intended to limit the scope of the invention. Reference should be made to the appended claims therefore rather than the specification as indicating the scope of the invention. 

What is claimed is:
 1. A curve generator for generating a smooth curve with a pair of circular arcs between a start point P₁ and an end point P₂ where the generated curve is tangent to given tangent lines M₁ and M₂ at said points P₁ and P₂, said curve generator comprising selecting means for selecting a point Q which is the smooth junction point of a pair of arcs on the locus satisfying the formula:

    ∠P.sub.1 QP.sub.2 =π-1/2(θ.sub.1 +θ.sub.2)

where θ₁ and θ₂ are the angles between the line P₁,P₂ and the tangent lines M₁ and M₂ respectively, decision means for deciding the revolutional direction of said arcs, and generating means coupled to said selecting means and said decision means for generating said two arcs.
 2. A curve generator according to claim 1 wherein the coordinates of the point Q(X_(q), Y_(q)) are obtained by the formula: ##EQU4## where X₁ and Y₁ are coordinates of the point P₁, X₂ and Y₂ are coordinates of the point P₂, m₁ =(M+tan α)/(1-Mtan α), m₂ =(M-tan β)/(1+Mtan β), tan β=(T-tan α)/(1+Ttan α), T=tan (θ₁ +θ₂)/2, M=Y₂ -Y₁ /X₂ -X₁, tan α is the parameter, and the revolutional direction of the arcs is defined according to the sign of D₁ =(M₁ -m₁)/(1+M₁ m₁) and D₂ =(M₂ -m₂)/(1+M₂ m₂).
 3. A curve generator according to claim 1 wherein said selecting means and said decision means includes digital calculators for calculating for the coordinates of the point Q and the sign of the values D₁ and D₂.
 4. A curve generator according to claim 3 wherein said digital calculators provide the data concerning two circular arcs in time sequence.
 5. A curve generator for generating a curve with a pair of circular arcs between the start point P₁ and the end point P₂ where the generated curve is tangent to given tangent lines M₁ and M₂ at said points P₁ and P₂ respectively, said curve generator comprising:(a) a P₁ register for storing the coordinates P₁ (X₁, Y₁); (b) a P₂ register for storing the coordinates P₂ (X₂, Y₂); (c) an M₁ register for storing the slope of the tangent to the curve at the starting point; (d) an M₂ register for storing the slope of the tangent to the curve at the end point; (e) a T register for storing the value T=tan (θ₁ +θ₂)/2 where θ₁ and θ₂ are angles between the line P₁ P₂ and said tangent lines; (f) a V register for storing the parameter V=tan α; (g) M calculator means coupled to said P₁ register and said P₂ register for calculating M=(Y₂ -Y₁)/(X₂ -X₁); (h) m₁ calculator means coupled to said V register, said M calculator means for calculating m₁ =(M+tan α)/(1-Mtan α); (i) m₂ calculator means coupled to said V register, said M calculator means and said T register for calculating m₂ =(M-tan β)/(1+Mtan β), where tan β=(T-v)/(1+Tv); (j) Q calculator means coupled to said m₁ and m₂ calculator means and said P₁ and P₂ registers for calculating ##EQU5## (k) D₁ calculator means coupled to said M₁ register and said m₁ calculator means, for calculating D₁ =(M₁ -m₁)/(1+M₁ m₁); (l) D₂ calculator means coupled to said M₂ register and said m₂ calculator means, for calculating D₂ =(M₂ -m₂)/(1+M₂ m₂); (m) first output means for providing the data concerning the first arc (P₁ (X₁, Y₁), Q(X_(q), Y_(q)), M₁ and the sign of D₁); (n) second output means for providing the data concerning the second arc (Q(X_(q), Y_(q)), P₂ (X₂, Y₂), M₂ and the sign of D₂). 